Average age of A and B is 30 years, that of B and C is 32 years and the average age of C and A is 34 years. The age of C is

Options

A. a. 33 years
B. b. 34 years
C. c. 35 years
D. d. 36 years

Answer

To find the age of C based on the given average ages of A, B, and C, we can follow these steps:

### Step 1: Set up the equations based on the averages given in the question.

1. The average age of A and B is 30 years:
\[ \frac{A + B}{2} = 30 \implies A + B = 60 \quad \text{(Equation 1)} \]

2. The average age of B and C is 32 years:
\[ \frac{B + C}{2} = 32 \implies B + C = 64 \quad \text{(Equation 2)} \]

3. The average age of C and A is 34 years:
\[ \frac{C + A}{2} = 34 \implies C + A = 68 \quad \text{(Equation 3)} \]

### Step 2: Express B in terms of A using Equation 1.

From Equation 1:
\[ B = 60 - A \quad \text{(Equation 4)} \]

### Step 3: Substitute Equation 4 into Equation 2.

Substituting \(B\) from Equation 4 into Equation 2:
\[ (60 - A) + C = 64 \]
This simplifies to:
\[ C - A = 64 - 60 \implies C - A = 4 \implies C = A + 4 \quad \text{(Equation 5)} \]

### Step 4: Substitute Equation 5 into Equation 3.

Now substitute \(C\) from Equation 5 into Equation 3:
\[ (A + 4) + A = 68 \]
This simplifies to:
\[ 2A + 4 = 68 \]
Subtract 4 from both sides:
\[ 2A = 64 \implies A = 32 \]

### Step 5: Find the age of C using Equation 5.

Now substitute \(A\) back into Equation 5 to find \(C\):
\[ C = A + 4 = 32 + 4 = 36 \]

### Conclusion

The age of C is:
\[ \boxed{36} \text{ years} \]

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